where \(\zeta \)  is a criterion value usually around unity.
We note that a theoretical ambiguity of this criterion may arise due to the gauge dependence and uncertainty from higher-order calculations, as studied in Ref. [1]}. Nevertheless, we will still use it as a useful guidance to the relevant regions in the parameter space.
For the left-hand side (LHS) of Eq. (REF ), the ratio \(v_c/T_c\)  obtained by using our procedure provided below is gauge-dependent.
As mentioned above, the full one-loop effective potential is gauge-dependent due to the thermal corrections.
With the leading-order high-temperature expansion of the effective potential and the implementation of Nielsen's identity,
Ref. [1]} provided a gauge-independent determination of \(v_c\) , \(T_c\)  and thus \(v_c/T_c\) .
To estimate the impacts of gauge dependence in criterion (REF ), we will adopt the gauge-independent effective potential, which is obtained by truncating the one-loop effective potential at second order in the EW gauge couplings, to calculate
\(v_c/T_c\)  in section . We will show that in comparison with the full one-loop potential, using the gauge-independent potential generally reduces the critical temperature
while increasing \(v_c/T_c\) . Thus, the samples satisfying criterion (REF ) in the Landau gauge can also satisfy the gauge-independent version of criterion (REF ).
For the RHS of criterion (REF ), there exist several sources of theoretical uncertainties in obtaining this quantity (a summary of these uncertainties can be found in Ref. [1]} and references therein).
One of the uncertainties is the lower bound on the “washout factor,” \(S>e^{-X}\) , where \(S\)  is the ratio of the baryon densities after and before the phase transition. Taking \(X\simeq 10\)  could lead to the conventional criterion value, \(\zeta \simeq 1.0\) . However, for a certain scenario, uncertainties in the value of \(X\)  could arise from the efficiency of the CP violation mechanism and
the duration of phase transition. A more realistic treatment of the criterion is to replace unity by a range determined by an appropriate choice of \(X\)  and the other theoretical inputs [1]}.
In this work, we will simply take the conventionally used criterion value \(\zeta \simeq 1.0\)  for the determination of a sufficiently strong phase transition.
